In the book Partial Differential Equations by L.C. Evans there are two solution concepts for Hamilton Jacobi equations
A. VISCOSITY SOLUTION(which is defined in chapter 10)
B. WEAK SOLUTION(which is defined in chapter 3)
I have the following doubts....
1)Is the visocotiy solution lipschitz continuous? if so does it satatisfy the equation pointwise a.e.
2)Are these two solutions same?
if so what is the justification?
Definition of Viscosity solution(L.C.Evans, PDE, Chapter 10): Assume that $u$ is bounded and uniformly continuous on $R^n \times [0,T]$,for each $T\geq 0$. We say that u is viscosity solution of the initial value problem
$u_t+H(Du,x)=0$ in $R^n \times (0,\infty)$,
u=g on $R^n \times \{t=0\}$ provided
A)$u=g$ on $R^n \times (0,\infty)$, and
B)for each v $\in C^ \infty (R^n \times (0,\infty))$,
if $u-v$ has a local maximum at a point $(x_0,t_0) \in R^n \times (0,\infty)$ then $v_t(x_0,t_0)+H(Dv(x_0,t_0),x_0)\leq 0$
and if
if $u-v$ has a local minimum at a point $(x_0,t_0) \in R^n \times (0,\infty)$ then $v_t(x_0,t_0)+H(Dv(x_0,t_0),x_0)\geq 0$
Yes, wherever a viscosity solution is differentiable, it satisfies the PDE. In many cases the viscosity solution is Lipschitz (e.g., it is Lipschitz in the setting of Evans Chapter 10), but there are circumstances where the viscosity solution is less regular (continuous, or even discontinuous).
Viscosity solution is more general than weak solution. The weak solution from Chapter 3 is a viscosity solution, but the weak solution is required to be semi-concave, which requires $H$ to be convex. In more general settings (say, $H$ not convex), the notion of weak solution is not applicable, but we still have viscosity solution.